Limit, continuity and differentiability
Understand the Problem
The question is requesting information or clarification on the concepts of limit, continuity, and differentiability in the context of mathematical analysis. This involves understanding how functions behave as they approach a certain point (limits), whether a function is continuous at a point (continuity), and whether a function has a derivative at a point (differentiability).
Answer
Continuity is required for differentiability, but not vice versa.
A function must be continuous to be differentiable, but continuity does not imply differentiability.
Answer for screen readers
A function must be continuous to be differentiable, but continuity does not imply differentiability.
More Information
In calculus, understanding the relationship between continuity and differentiability is crucial. While a differentiable function is always continuous, not all continuous functions are differentiable, highlighting subtle differences, such as sharp corners, in function behavior.
Tips
A common mistake is assuming that if a function is continuous everywhere, it is also differentiable everywhere. Points such as cusps or corners can make a function non-differentiable at that point.
Sources
- Limits and Continuity: A function must have a limit at a point to be continuous there... - geeksforgeeks.org
- 1.7: Limits, Continuity, and Differentiability - Mathematics LibreTexts - math.libretexts.org
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